CoCalc Public Fileswww / talks / harvard-talk-2004-12-08 / e2heights / mazur_request.txtOpen with one click!
Author: William A. Stein
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Karl Rubin and I were wondering whether you could do a p-adic height
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computation, the point of which is to figure out whether the p-adic
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regulator of elliptic curves has a "tendency" to be not divisible by p. We
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want to avoid a few degenerate cases, so the smallest p that is useful to
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us is p=5. Is the strategy-- described below--, that systematically runs
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through 5-adic height computations for points on quadratics twists of the
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(non-3-Eisenstein) elliptic curve of conductor 37 reasonably easy to do?
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Our untwisted equation is E: y^2 = 4x^3-4x+1. For any nonzero integer D we
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consider the twisted elliptic curve E_D: D.y^2 = 4x^3-4x+1. Put F(x): =
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4x^3-4x+1. Let x run through integers
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a= 0, \pm 1, \pm 2, \pm3, ...
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and for each of these a's,
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1) Factor F(a) = d.b^2, where d is square-free;
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2) throw out the ones where d is divisible by 5, or where the parity of the
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rank of E_d is even;
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3) for the rest, viewing P:=(a,b) as a rational point-- it will be of
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infinite order--on the elliptic curve E_d, so one can compute the 5-adic
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height of P on the elliptic curve E_d: this should be a 5-adic integer if
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the height is normalized right, and there should be a good number of
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instances where this height is a 5-adic unit; throw them out, and
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4) print the rest.
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Note that we only want to know the 5-adic height modulo 5.
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Then it would be our job to figure out whether strange things occur for
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this printed list of instances: e.g., the point P might be divisible by 5,
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or the Mordell-Weil rank might be > 2, or other stuff..., and we want to
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know what other stuff there might be.
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